K11n148

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K11n147.gif

K11n147

K11n149.gif

K11n149

Contents

K11n148.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n148 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X14,4,15,3 X5,19,6,18 X7,20,8,21 X9,1,10,22 X11,6,12,7 X2,14,3,13 X15,9,16,8 X17,10,18,11 X19,13,20,12 X21,16,22,17
Gauss code 1, -7, 2, -1, -3, 6, -4, 8, -5, 9, -6, 10, 7, -2, -8, 11, -9, 3, -10, 4, -11, 5
Dowker-Thistlethwaite code 4 14 -18 -20 -22 -6 2 -8 -10 -12 -16
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11n148 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2,3\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n148/ThurstonBennequinNumber
Hyperbolic Volume 15.4617
A-Polynomial See Data:K11n148/A-polynomial

[edit Notes for K11n148's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -2

[edit Notes for K11n148's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-10 t^2+14 t-15+14 t^{-1} -10 t^{-2} +5 t^{-3} - t^{-4}
Conway polynomial -z^8-3 z^6+3 z^2+1
2nd Alexander ideal (db, data sources) \left\{5,t^2+2 t+1\right\}
Determinant and Signature { 75, 2 }
Jones polynomial -2 q^6+5 q^5-9 q^4+12 q^3-12 q^2+13 q-10+7 q^{-1} -4 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +z^6-7 z^4 a^{-2} +4 z^4 a^{-4} +3 z^4-z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} +z^2+3 a^{-2} - a^{-6} -1
Kauffman polynomial (db, data sources) 3 z^9 a^{-1} +3 z^9 a^{-3} +12 z^8 a^{-2} +6 z^8 a^{-4} +6 z^8+4 a z^7-z^7 a^{-1} -z^7 a^{-3} +4 z^7 a^{-5} +a^2 z^6-36 z^6 a^{-2} -16 z^6 a^{-4} +z^6 a^{-6} -18 z^6-11 a z^5-15 z^5 a^{-1} -10 z^5 a^{-3} -6 z^5 a^{-5} -2 a^2 z^4+33 z^4 a^{-2} +21 z^4 a^{-4} +4 z^4 a^{-6} +14 z^4+6 a z^3+12 z^3 a^{-1} +12 z^3 a^{-3} +9 z^3 a^{-5} +3 z^3 a^{-7} -8 z^2 a^{-2} -8 z^2 a^{-4} -4 z^2 a^{-6} -4 z^2-z a^{-1} -2 z a^{-3} -4 z a^{-5} -3 z a^{-7} -3 a^{-2} + a^{-6} -1
The A2 invariant q^8-2 q^6+q^4-2 q^2+2 q^{-2} - q^{-4} +6 q^{-6} - q^{-8} +3 q^{-10} - q^{-12} -2 q^{-14} + q^{-16} -2 q^{-18} + q^{-20} - q^{-22}
The G2 invariant Data:K11n148/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a223,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n168,}

Vassiliev invariants

V2 and V3: (3, 4)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
12 32 72 158 34 384 \frac{2048}{3} \frac{320}{3} 128 288 512 1896 408 \frac{32911}{10} -\frac{462}{5} \frac{23102}{15} \frac{401}{6} \frac{2351}{10}

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=2 is the signature of K11n148. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-1012345χ
13         2-2
11        3 3
9       62 -4
7      63  3
5     66   0
3    76    1
1   47     3
-1  36      -3
-3 14       3
-5 3        -3
-71         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}_2^{2} {\mathbb Z}^{2}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

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K11n147.gif

K11n147

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K11n149